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1 Oll Geeralized Berstei Polyomials i CAGD Guido Walz Nr.86 November 1988 'i-'.,,~~ :'>'-. "',.,- ~. ~,..._.. w. ",... -i. _. ",.' ",...,.,.'., ' ;.' ~-.,."""",:.... _...~...'-.... _,,

2 O Geeralized Berstei PolYllorials i CAGD Guido Walz Abstract: A cetral topic i CAGD is the represetatio of curves ad surfaces by, polyomial iterpolatio operators, i particular by Be'1stei polyomials. I this paper we preset two differet types of geeralized Berstei polyomials. The first oe goes back to a idea of D..Stacu [5, 6, 7]; here a dass of polyomials qv is defied, which deped cotiuously o a additioal parameter a. For a = 0, the qv coicide with the ordiary Berstei polyomials, whereas for a = -1/ they are idetical with the Lagrage polyomials. A secod type of geeralizatio is essetially due to G.G.Loretz [3]; here, the Berstei polyomials are defied with respect to a geeralizedpolyomial space, cosistig of fuctios of the form L:~=o al/xo" for some 0 = ao ::;... ::; a. Very importat for applicatio i CAGD is the fact that the "ice" properties of the Berstei polyomials, such as positivity, partitio of uity ad the recursive computability carry over to these geeralizatios. Keywords: Berstei Polyomials, Stacu Operators, Curve Fittig.

3 Geeralized Berstei Polyomials 2 1. Itroductio Very importat tools i OAGD are so-called liear iterpolatio operators for the represetatio of curves i IR2 or IR3 ad also - ii oe uses for example tesor product methods - of suhaces i IR3. Here, followig DeVore [2, p.26], a operator L is called liear iterpolatio operator, if it is of the form L(J, x) = L f(xv)hv(x), v=o where f deotes a certai prescribed fuctio, ad the Xo,, X as weil as the variable x are real umbers, usually restricted to the iterval [0,1]. It is ot meat that L really iteipolates the fuctio f i the poits Xv ; the term iterpolatio oly idicates that the shape of L(J, ) depeds oly o a fiite umber of values f (xv) - i cotrast, for, example - to itegral operators. i' For applicatio i OAGD, oe replaces the f(xv) by the cotrol poits b v, where bv E IR2 or IR3, v = 0,...,, ad the resultig L(x) = L bvhv(x) 1=0 represets a curve which depeds oly o the cotrolpoits b v (cf., also for the followig, the excellet survey paper of Böhm, Fari ad Kahma [1]). polyomials The two most importat choices for the basis fuctios h v are the Lagrage lv (x) ad the Berstei (basis) polyomials Pv(x) = (~)xv(1-x)-v for x E [0,1]. I particular the latter o es are of great iterest; the theoretical back. groud is give by the weu-kow theorem of Berstei ad Korovki, which says that for fe 0[0, 1J the pol~'omial operators B (J, x) = L f(~)pv(x) v=o v

4 Geeralized Berstei Polyomials 3 coverge to f,as teds to ifiity. The crucial poit for the applicabijity of the correspodig operators R (a') = I: bvpv (x) 1=0 (1.1) i OAGD are the followig rour properties: (1) (2) (3) (4) Pl/(X) ~ 0 for xe [0,1], I: Pv(X) == 1, 1=0 For EIN ad 1 ~ v ~ - 1 : Pv(X) = (1 - x) P-l,v(X) + X P-l,v-I(X), J (1.2) where i particular (4) is importal;, because it leads to the developmet of de Casteljau type algorithms; (3) guaratees that two operators ofthe type (1.1) ca be cotiuously, coected.,. I this paper we preset two differet geeralizatiosof the Berstei polyomi. als, whieh are both suited to improve the ßexibility of curve represetatio methods i CAGD. The first oe goes back to a idea of D..Staeu 151 ad will be preseted i the ext sectio; a dass of polyomials q"v is defied, which deped o just oe additioal real parameter, say a, ad cover, for example, the Lagrage as weil as the Berstei polyo' mials as special cases. To our believe, this cocept will be of great iterest of OAGD systems. for desigers I sectio 3, we preset aother type of geeralizatio, which is due to G.G.Loretz [3]; here the basic idea is to use Berstei polyomials, which are defied with respect to the space {xao,...,x a ",}, O=ao <... <aeir,isteadofthespace TI' It is importat that both types of geeralizatios preserve the properlies (1.2) of the Berstei operators.

5 Geeralized Berstei Polyomials 4 2. Tlle Stae Polyomials ad Operators As it is kow the Berstei operators (1.1) are quit~ suitable for the represe. tatio of curves, due to their ruce properties (1.2); furtheore, they do ot ted to a oscillatig curve, as icreases - i cotrast, for example, to Lagrage polyomials. O the other had, oe sometimes would like to have (polyomial) operators which react a little bit more sesitive o alteratios of the cotrol poits b ll ; ad, eve more, that the degree of this ifluece ca be cotrolled by a user-defied parameter. Such a possibility is give by usig the followig operators, which were itroduced by D.D.Stacu i [5] ad frther ivestigated by the same author i [6, 7] ad G.Mühlbach _ [4]: Defiitio 2.1: Let x ad a be arbitrary real uib~rs ad defie for v E IN o : j' 11-1 \Oo(x,a):=l, \01I(x,a):= II(x+,\a), ad ).=0 ( )._ () )Oll(x, a). )O-lI(l- x, a) qll X, a.- ()' EIN. v )O 1, a The for V = 0, 1,..., the qll are polyomials (i x) of degree, which deped o the parameter a (assurig \O (1, a) 1= 0); they will be deoted as Stacu polyomials. The correspodig operators will be called Stacu operators. Q(X, a).- I: bllqll 11=0 (x, a) (2.1 ) Their mai properties, which shold be compared to (1.2), are summarized i terms of the followig theorem (cf. [4]): Theorem 2.2: (1) For x EX, qv (x, a) ~ 0, where if a ~ 0, { X = [a(l - ), 1- a(l - )], if a < O. X = [0,1],

6 Geeralized Berstei Polyomials 5 (2) For fixed a, L qv(x,a) = 1. v=o (3) Q(O,a) = bo ad Q(1,a) = b (4) For EIN ad 1 ~ v ~ - 1 : qv (x, a) = 1. ( ). ((1- x + ( - 1/ -1) a) q-l v(x, a) + (x + (1/ -1) a) q-l v-dx, all. l+-1a ', Proof: The properties (1), (2) ad (4) ca already be foud i G.Mühlbach's paper [4]; the proof of (3), however, ca be doe by straightforward calculatios, usig the fact that, Now is is iterestig to aalyse the depedece of Qh(x, a) o the parameter a: Stacu already poited out that qv (x, a) coicides with the Berstei polyomial Pv(x), if a = 0, ad with the Lagrage polyomial lv(x), if a = -ll, where i the latter case the Xv must be chose equal to v I [6]. Now we have a closer look oto the behaviour of Q(X, a} for several values of a; first, if a< -ll, we have a highly oscillatig curve, which has - to our believe - o practical applicatios. Much more iterestig is the rage -ll ~ a ~ 0; here we get operators, which lie "betwee" the Lagrage ad the Berstei operators (ad deped cotiuously o the parameter a). ]\:[ore precisely, the Q behave as foliows: startig, for a = 0, with the well-kow Berstei operators, oe gets, as a teds to -ti, operators, which become more ad more iterpolatory with respect to the cotrol poits b v, util oe eds up at the Lagrage operators, which do i fact iterpolate the b v Figure 1 sketches this behaviour for = 3 ad a= 0, -1/3, -2/3, -ll.

7 Geeralized Berstei Polyomials 6 Figure 1: Qa(x, a) for various values of a Fially we show that for a -;. +00, the resultig curves ted to a straigth lie: Lemma 2.3: the For fixed x E [0,1], let Q(x, a) deote the Stacu operator (2.1); lim Q(x,a) - (l-x)bo+xb a... += Proof: First we ote that l1-1 (1 - x +,\a) qo (x, a) = ( ).\=0 l+,\a Therefore ad, as it ca be show i a completely (l-x)( - I)! a O(a - 2 ) ( - I)! a O(a - 2 ) lim qo (x, a) = 1 - x, a... +oo aalogous maer, for a -;. 00 lim q(x,a) = X a~+oo So we are left to prove that lim qv(x,a) - 0 for1<v<. a--++oo

8 Geeralized Berstei Polyomials 7 But this follows at oee, beeause -0 for a L.l So, roughly spake, the shape of the curves defied by the Stacu operators (2.1) varies rom a highly oseillatig polyomial eurve to a straight lie, as a moves from -00 to +00; the most iterestig rage for CAGD applicatios is give b~, -1/ ~ a ~ o. For example, oe could leave the defiitio of the parameter a free to the user of the CAGD system; ifhe wats to costruct a curve (01' surface), which depeds stroger o the cotrol poits, he would have to choose a value ear -l/. If, o the other had, a curve is wated whieh is ot so sesitive agaist moves the cotrol poits, a value of a ear 0 would be suited. From trus poit of view, oe could deote a as the "sesitivity parameter" of the system. 3. The Loretz Polyomials ad Operators I the followig, we sketch a completely differet type of geeralized Berstei polyomials, which is due to G.G.Loretz.We restrict ourselves to a very short prese. tatio of the mathematieal backgroud ad refer the iterested reader to Loretz' book [3]. Let 0'0 = 0 ad av ~ 0 be arbitraryad xe [0, I}. Defie, for EIN, P(X).- 1, ad for v = 0,..., - 1 : 1 f X Z dz 2' c (z-o'v).. ' z-a) Pv(X) := (_1)-v av+l... a. -. (' (3.1) where C is a simply dosed curve i the complexplae such that all the poits 0'1,"" a lie i the iterior of C.

9 Geeralized Berstei PolyoIiiials 8 If the a/l are mutually differet, the (3.2) I particular, if aj.l = 11 for 11 = 0,...,, Pv coicides with the ordiary Berstei basis polyomials Pll' I this sese the Pv ca be deoted as geeralized Berstei polyomials. For completeess we treat also the case that some of the aj.l coicide; the P/l is a liear combiatio of fuctios of the form where. kj.l deotes the multiplicity of ajl It ca be show (cf. [3, sect. 2.71) that the properties/(1.2) carry over also to this type of geeralizatio. For umbers (1) through (3) this is true for all 0 = ao :$... :$ a. For example, usig the idetity 1 1 a al... a z z-a(z-a-i)(z-a). (z-ao)"'(z-a) (ote that ao = 0) ad the defiitio (3.1), oe proves,that 1 1 z L Pll(X) = ~ ~dz = 1. 11=0 ~z C z The validity of recursio formulas like (1.2), (4) caot be proved i geeral, but for some special choices of the expoets ajl For example, i the importat case that for some pe IR, p > 0, oe easily shows - e.g. usig (3.2) - that Pv (x) =(1 - x P ) P-l,v(X) + xp P-l,v-dx) for EIN ad 1:$ v :$ - 1.

10 Geeralized Berstei Polyomials 9 Refereces [1] W.Böhm, G.Fari ad J.Kahma: A Surey o! Curve ad Surface Methods i CA GD. Computer Aided Geometrie Desig 1 (1984), 1-60 [2] R.A.DeVore: The Approximatio o! Cotiuous Fuctios by Positive Liear Operators (Leeture Notes i Mathematies 293). Spriger, Berli/Reidelberg/N ew York 1972 [3] G.G.Loretz: Berstei Polyomials. Uiversity of Toroto Press, Toroto [4] G.Mühlbaeh: Verallgemeieruge der Berstei- ud der Lagrage-Polyome. Rev. RoumaieMath. Pures Appl. 1.5 (1970), [5] D.D.Staeu: O a New Positive Liear Polyomial Operator. Proc. Japa Aead. 44 (1968), [6] D.D.Stacu: Approximatio o! FUllctios bya New 01ass o! Polyomial Operators..~ Rev. Roumaie Math. Pures Appl. 13 (1968), [7] D.D.Stacu: Use of Probabilistie Methods i the Theory of Uiform Approximatio of Cotiuous Fuctios. Rev. Roumaie Math. Pures Appl. 14 (1969), Dr. Guido Walz Fakultät für Mathematik ud Iformatik Uiversität Maheim D-6800 J\1ANNREIM 1 West Germay